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The extremal function associated to intrinsic norms. (English) Zbl 1036.32026

Let \(\Omega= \Omega'\setminus \bigcup^N_{k=1} \overline\Omega_k\) where \(\Omega'\) and \(\Omega_k\) are bounded pseudoconvex domains in \(\mathbb{C}^n\) with smooth boundary, \(\bigcup^N_{k=1} \overline\Omega_k\) is holomorphically convex in \(\Omega'\) and \(\overline\Omega_1,\dots, \overline\Omega_N\) are pairwise disjoint.
Associated to some variational problem on \(\Omega\), E. Bedford and B. Taylor proved in [Am. J. Math. 101, 1131–1166 (1979; Zbl 0446.35025)] the existence of an extremal function \(u\) which is plurisubharmonic on \(\Omega\), Lipschitz on \(\overline\Omega\) and satisfies the Monge-Ampère equation \((dd^c u)^n= 0\) with boundary conditions \(u|_{\partial\Omega'}= 1\), \(u|_{\partial\Omega_k}= 0\).
In this paper the author proves that the extremal function \(u\) is the limit of a sequence of smooth plurisubharmonic functions \(u_k\) with \(\| u_k\|_{C^{1,1}(\Omega)}\) bounded. He then concludes that \(u\) has \(C^{1,1}\) regularity and proves some results obtained by E. Bedford and B. Taylor [loc. cit.] under the assumption that \(u\) is \(C^2\).

MSC:

32W20 Complex Monge-Ampère operators
32U35 Plurisubharmonic extremal functions, pluricomplex Green functions
31C10 Pluriharmonic and plurisubharmonic functions

Citations:

Zbl 0446.35025
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